Term | Description |
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(1) Identification of system status by E-GPS states | |
(a) E-GPS state | It is a convex set \(R_s\subseteq \mathcal{D}_{g\phi e}\) with all its elements sharing the same status type, e.g., the state \(s_{11}\) in Fig. 1f, and the probability that the system visits this state (i.e., within \(R_s\)) is bigger than a threshold, i.e., the state is not rare. Empirically, the percentage of a given set of samples falling in \(R_s\) should be larger enough |
(b) Prob. E-GPS state (d-state vs. c-state ) | It is a state that is not rare but prob. (probabilistic) in a sense that each element in \(R_s\) is either Type ‘+’ in a number \(n_S^{+}\) or Type ‘−’ in a number \(n_S^{-}\), in two categories: d-state (Dedicated state): \(max\{n_S^{+},n_S^{-}\}\) is significantly bigger than \(min\{n_S^{+},n_S^{-}\}\). Empirically, samples falling in \(R_s\) are mostly dedicated to a same status type, e.g., the state \(s_{00}\) in Fig. 1f. c-state (Confusing state); otherwise, i.e., two status types compete samples in \(R_s\), e.g., the states \(s_{10}\) and \(s_{01}\) in Fig. 1f |
(c) c-state (cuttable vs noncuttable) | It is a c-state with at least one convex subset that is able to be cut off as a d-state, e.g., the state \(s_{01}\) in Fig. 1f; otherwise the c-state is said to be noncuttable under the current settings of \(\mathcal{D}_{g\phi e}\), e.g., the state \(s_{10}\) in Fig. 1f |
(d) Learning configuration of states | Overall, a set of at least one d-state and c-states (if any) is learned from a given set of samples, featured by not only these states but also their configuration that encodes the locations and mutual relations of these states, as illustrated in Fig. 1h |
(2) Refinements of E-GPS states | |
(a) cutting | Cut a cuttable c-state by linear separation, e.g., SVM (Suykens and Vandewalle 1999; Suykens et al. 2002) or FDA by Eqs. (11) and (12) in Ref. Xu (2015a), via refining condition, e.g., the red line cuts \(s_{01}\) in Fig. 2f, which results in one convex subset as a d-state and one size-reduced c-state that may be still cuttable c-state |
(b) merging | Merge adjacent d-states if their union is still convex, e.g., merging \(s_2, s_3, s_4\) in Fig. 2f. In addition, merge adjacent c-states, e.g., \(s_5, s_6\) in Fig. 2f |
(c) growing | Grow each d-state s with \(Type(s)=\)‘green’ by including those adjacent ’green’ samples if the enlarged subset is still convex, and also grow each d-state s with \(Type(s)=\)‘red’ by including those adjacent ’red’ samples if the enlarged subset is still convex |
(d) treating | Use additional conditions (e.g., one more variable is added to \(\mathbf{\phi }\)) such that more ’green’ samples in the c-states become adjacent to and able to be re-allocated into some d-states in the above ways |
(3) Conditional phenotype analyses based on E-GPS states | |
(a) analysis per d-state | Prognosis analyses test whether \(max\{n_S^{+},n_S^{-}\}\) differs from \(min\{n_S^{+},n_S^{-}\}\) significantly by \(\chi ^2\) test or Fisher exact test to identify whether this state is good for prognosis, while the boundary of this state indicates the conditions under which the judgement is made. Moreover, prognosis of a unlabelled sample may be made by an one-class classifier obtained from these conditions |
Survival analyses plot K-M curves on samples with survival record and make the log rank test or the Cox proportional hazards test | |
Subtype analyses stratify samples of this state into each subtype, test the enrichment of each subtype in this state, plot K-M curves on each stratification, and examine the correlation or the intersection of each subtype to good and bad prognosis, as shown in Fig. 1h | |
(b) analysis cross d-states | Differentiation test on whether there is a significant difference pair-wisely either between samples of different d-states or between samples associated with different values of a phenotype, in one of the following manners: |
\(*\) A t-test when we ignore e and merely consider a univariate g; | |
\(*\) A multivariate test, e.g., Hotelling test Hotelling (1931), BBT test [see Table 6 in Ref. Xu (2015a)], and property-oriented test [see Algorithm 1 in Ref. Xu (2016)]; | |
\(*\) Model-based test proposed by Eqs. (29–31) in Ref. Xu (2015a); | |
\(*\) Logistic- or Cox-regression. On the lefthand of \(\eta (\phi _t)=\mathbf{b}^T g_t+ \mathbf{a}^T e_t +c+\varepsilon _t\), we test whether one or more of coefficients of \(\mathbf{b}\) are zero and whether one or more of coefficients of \(\mathbf{a}\) (e.g., by the score test or the Wald test) to examine whether the corresponding variables take roles significantly | |
Staging that is related to subtypes but different, staging involves subtypes in a temporal order. The later stage is usually more serious than the earlier stage, which may be learned via the transfer probabilities \(p(s_i|s_j)\) cross the states in Fig. 1h | |
Cross-state integration by comparing the configuration of states to enhance the differentiation study above. Moreover, cross-state combination can further provide better performance, as illustrated in Fig. 1h. Given the output measure \(\zeta _{j,t}\) (e.g., p value, classification error, and predicted regression) for a particular sample t, we may get one weighted average \(\zeta _t=\sum _j \zeta _{j,t} p(s_j|t)\), as well as a combined classification rule \(p(+|t)=\sum _j p(+|s_j) p(s_j|t)> p(-|t)=\sum _j p(-|s_j) p(s_j|t)\) |